On the extreme points of sets of absolulely separable and PPT states
arXiv:2409.14347 · doi:10.1103/7pnp-qgrq
Abstract
The absolutely separable (resp. PPT) states remain separable (resp. positive partial transpose) under any global unitary operation. We present a compact form of the extreme points in the sets of absolutely separable states and PPT states in two-qubit and qubit-qudit systems. The results imply that each extreme point has at most three distinct eigenvalues. We establish a necessary and sufficient condition for determining extreme points of the set of absolutely PPT states in two-qutrit and qutrit-qudit systems, expressed as solvable linear equations. We also demonstrate that any extreme point in qutrit-qudit system has at most seven distinct eigenvalues. We introduce the concept of robustness of nonabsolute separability. It quantifies the minimal amount by which a state needs to mix with other states such that the overall state is absolutely separable. We show that the robustness satisfies positivity, invariance under unitary transformation, monotonicity and convexity, so it is a good measure within the resource theory of nonabsolute separability. Analytical expressions for this measure are given for pure states in arbitrary system and rank-two mixed states in two-qubit system.
arXiv comment22 pages, 1 figure
References in corpus (43)
- Quantum entanglement
- Quantum Cryptography
- Separability Criterion for Density Matrices
- A computable measure of entanglement
- Separability of Mixed States: Necessary and Sufficient Conditions
- Entanglement detection
- On the volume of the set of mixed entangled states
- Separability criterion and inseparable mixed states with positive partial transposition
- Quantum Resource Theories
- Robustness of entanglement
- Direct estimations of linear and non-linear functionals of a quantum state
- Universal Quantum Estimator
- Randomizing quantum states: Constructions and applications
- Largest separable balls around the maximally mixed bipartite quantum state
- Maximally entangled mixed states of two qubits
- Geometry of entangled states
- Bounds on Multipartite Entangled Orthogonal State Discrimination Using Local Operations and Classical Communication
- How robust is a quantum gate in the presence of noise?
- Genralized Robustness of Entanglement
- Hilbert--Schmidt volume of the set of mixed quantum states
- Entanglement of multiparty stabilizer, symmetric, and antisymmetric states
- Separability from Spectrum for Qubit-Qudit States
- PPT from spectra
- Estimating entanglement monotones with a generalization of the Wootters formula
- Additivity and non-additivity of multipartite entanglement measures
- Connecting the generalized robustness and the geometric measure of entanglement
- Better bound on the exponent of the radius of the multipartite separable ball
- Extreme points of the set of density matrices with positive partial transpose
- Witness of mixed separable states useful for entanglement creation
- Determining eigenvalues of a density matrix with minimal information in a single experimental setting
- Entangled states close to the maximally mixed state
- Cones of ball-ball separable elements
- Characterizing the boundary of the set of absolutely separable states and their generation via noisy environments
- The Inverse Eigenvalue Problem for Entanglement Witnesses
- Absolutely separating quantum maps and channels
- Eigenvalues, Separability and Absolute Separability of Two-Qubit States
- Invariance of separability probability over reduced states in 4x4 bipartite systems
- Positive reduction from spectra
- Boundary of the set of separable states
- Absolute separability witnesses for symmetric multiqubit states
- Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace
- Resource Theory of Non-absolute Separability
- Quasirandom estimations of two-qubit operator-monotone-based separability probabilities